Finite Group Databasebeta

← #36525573

Subgroups — #36525573

Order 22784.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030324C211yesmaximal, characteristic
2#26029270C211yesmaximal, characteristic
3#26029199C211yesmaximal, characteristic, complemented
4#26019156C211yesmaximal, characteristic, complemented
5#26019144C211yesmaximal, characteristic, complemented
6#26019132C211yesmaximal, characteristic, complemented
7#26019125C211yesmaximal, characteristic, complemented
8#26019104C211yesmaximal, characteristic, complemented
9#26019097C211yesmaximal, characteristic, complemented
10#26019072C211yesmaximal, characteristic, complemented
11C8×Dic356C211yesmaximal, characteristic, complemented
12#26019037C211yesmaximal, characteristic, complemented
13#26018726C211yesmaximal, characteristic, complemented
14#26018640C211yesmaximal, characteristic
15C89×#1453C211yesmaximal, characteristic, complemented
16#19152155C2211yescharacteristic
17C4×#13083574C2211yescharacteristic
18#19151940C411yescharacteristic
19#19151940C411yescharacteristic
20#19151884C411yescharacteristic
21#19151879C411yescharacteristic
22#19151809C411yescharacteristic
23#19151804C411yescharacteristic
24#19151762C411yescharacteristic
25#19151762C411yescharacteristic
26#19151728C2211yescharacteristic
27#19151728C2211yescharacteristic
28#19151726—21no—
29#19151726—21no—
30C8×Dic178C2211yescharacteristic
31C8×Dic178C2211yescharacteristic
32C8×Dic178C2211yescharacteristic
33C8×Dic178C2211yescharacteristic
34C4×Dic356C2211yescharacteristic
35C4×Dic356C2211yescharacteristic
36C4×#13083563C2211yescharacteristic
37C4×#13083563C2211yescharacteristic
38#19151677C2211yescharacteristic
39#19151677C2211yescharacteristic
40#19151674—21no—
41#19151674—21no—
42#19151655C2211yescharacteristic
43#19151653—21no—
44#19151626—21no—
45C2×C4⋊#10623494C2211yescharacteristic
46#19151556—21no—
47#19151550—21no—
48#19151351C2211yescharacteristic
49#19151351C2211yescharacteristic
50#19151349C2211yescharacteristic
51#19151349C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151330C2211yescharacteristic
55#19151330C2211yescharacteristic
56#19151330C2211yescharacteristic
57#19151330C2211yescharacteristic
58#19151327C2211yescharacteristic
59#19151327C2211yescharacteristic
60#431×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62C4×M16×C89C2211yescharacteristic
63#322×C89C2211yescharacteristic
64#322×C89C2211yescharacteristic
65#322×C89C2211yescharacteristic
66C82×C89C2211yescharacteristic
67C2×#13083574C2×C411yescharacteristic
68C2×Dic356C2×C411yescharacteristic
69C2×Dic356C2×C411yescharacteristic
70C2×#13083563C2×C411yescharacteristic
71C2×#13083563C2×C411yescharacteristic
72C2×#13083557—21no—
73C2×#13083557—21no—
74C22×#10623494—21no—
75#15668012C2×C411yescharacteristic
76#15668012C2×C411yescharacteristic
77#15668004—21no—
78#15667991—21no—
79#15667991C2×C411yescharacteristic
80#15667991C2×C411yescharacteristic
81C4×Dic178—21no—
82C4×Dic178C2×C411yescharacteristic
83C4×Dic178C2311yescharacteristic
84C4×Dic178C2311yescharacteristic
85C4×Dic178C2311yescharacteristic
86C4×D356C2×C411yescharacteristic
87C4×D356C2311yescharacteristic
88C4×C22⋊D89—21no—
89C4×C22⋊D89—21no—
90#15667971—21no—
91#15667971C2×C411yescharacteristic
92#15667969—21no—
93C42×D89—21no—
94#15667946C2×C411yescharacteristic
95#15667946C2×C411yescharacteristic
96#15667944C2×C411yescharacteristic
97#15667944C2×C411yescharacteristic
98C4⋊#10623494—21no—
99C4⋊#10623494C2311yescharacteristic
100C4⋊#10623494C2311yescharacteristic
101C4⋊#10623494C2311yescharacteristic
102C4⋊#10623494C2311yescharacteristic
103#15667941—21no—
104#15667941—21no—
105#15667941—21no—
106#15667941—21no—
107#15667941—21no—
108#15667941—21no—
109#15667937C2×C411yescharacteristic
110#15667937C2×C411yescharacteristic
111#15667937C2×C411yescharacteristic
112#15667937C2×C411yescharacteristic
113#15667937C2×C411yescharacteristic
114#15667937C2×C411yescharacteristic
115#15667934C2×C411yescharacteristic
116#15667934C2×C411yescharacteristic
117C22⋊#10623494—21no—
118#15667924—21no—
119#15667924—21no—
120#15667923—21no—
121#15667922—21no—
122#15667922—21no—
123C8×Dic89—21no—
124C8×Dic89—21no—
125C8×Dic89—21no—
126C8×Dic89—21no—
127C2×M16×C89D411yescharacteristic
128C2×M16×C89D411yescharacteristic
129C2×C42×C89C2311yescharacteristic
130C4⋊C8×C89—21no—
131C4⋊C8×C89C2×C411yescharacteristic
132C4⋊C8×C89C2×C411yescharacteristic
133C22⋊C8×C89—21no—
134C22⋊C8×C89C2×C411yescharacteristic
135C22⋊C8×C89C2×C411yescharacteristic
136#97×C89D411yescharacteristic
137#97×C89D411yescharacteristic
138C4×C8×C89C2311yescharacteristic
139C4×C8×C89C2311yescharacteristic
140C4×C8×C89C2311yescharacteristic
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144#13083574—22no—
145#13083574—41no—
146C2×Dic178—21no—
147C2×Dic178C22×C411yescharacteristic
148C2×Dic178C22×C411yescharacteristic
149C2×Dic178C22×C411yescharacteristic
150C2×D356C22×C411yescharacteristic
151C2×C22⋊D89—21no—
152C2×C4×D89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155Dic356—22no—
156Dic356—22no—
157#13083563—41no—
158#13083563—41no—
159#13083557—42no—
160#13083557—42no—
161C2×#10623494—21no—
162C2×#10623494—21no—
163C2×#10623494—21no—
164C2×#10623494—21no—
165C2×#10623494—42no—
166C4⋊Dic89—21no—
167C4⋊Dic89—21no—
168C4⋊Dic89—21no—
169C4⋊Dic89C22×C411yescharacteristic
170C4⋊Dic89C22×C411yescharacteristic
171C4⋊Dic89C22×C411yescharacteristic
172C4⋊Dic89C22×C411yescharacteristic
173#13083553—21no—
174#13083553—21no—
175#13083553—21no—
176#13083553—21no—
177#13083553—21no—
178#13083553—21no—
179C22⋊Dic89—21no—
180C22⋊Dic89—21no—
181C22⋊Dic89—21no—
182#13083551—21no—
183#13083551—21no—
184C4×Dic89—21no—
185C4×Dic89—21no—
186C4×Dic89—21no—
187C4×Dic89—21no—
188C4×Dic89—21no—
189C22×C4×C89C22×C411yescharacteristic
190C22×C4×C89C2×D411yescharacteristic
191C22×C4×C89#4111yescharacteristic
192M16×C89—22no—
193M16×C89—41no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C2×C8×C89C22×C411yescharacteristic
197C2×C8×C89C22×C411yescharacteristic
198C2×C8×C89C22×C411yescharacteristic
199C2×C8×C89C22×C411yescharacteristic
200C2×C8×C89C2×D411yescharacteristic
201C2×C8×C89C2×D411yescharacteristic
202C2×C8×C89C2×D411yescharacteristic
203C2×C8×C89C2×D411yescharacteristic
204C42×C89C22×C411yescharacteristic
205C42×C89C2×D411yescharacteristic
206C42×C89#4111yescharacteristic
207C42×C89C2411yescharacteristic
208C23×C89C4×D411yescharacteristic
209C22×D89—21no—
210C2×C4×C89—21no—
211C2×C4×C89—21no—
212C2×C4×C89—21no—
213C2×C4×C89—21no—
214C2×C4×C89—41no—
215C2×C4×C89C4×D411yescharacteristic
216C2×C4×C89C4×D411yescharacteristic
217C2×C4×C89C4×D411yescharacteristic
218C2×C4×C89C23×C411yescharacteristic
219C2×C4×C89C22×D411yescharacteristic
220C2×C4×C89C2×#4111yescharacteristic
221C22⋊D89—42no—
222C2×Dic89—21no—
223C2×Dic89—21no—
224C2×Dic89—21no—
225C2×Dic89—21no—
226C2×Dic89—21no—
227C2×Dic89—21no—
228C2×Dic89—21no—
229D356—21no—
230D356—21no—
231C4×D89—42no—
232C4×D89—41no—
233C4×D89—41no—
234C4×D89—81no—
235Dic178—21no—
236Dic178—21no—
237Dic178—21no—
238Dic178—21no—
239Dic178—21no—
240Dic178—81no—
241Dic178#13112yes—
242C8×C89—21no—
243C8×C89—21no—
244C8×C89—21no—
245C8×C89—21no—
246C8×C89—21no—
247C8×C89—21no—
248C8×C89—21no—
249C8×C89—41no—
250C8×C89—41no—
251C8×C89#13712yes—
252#10623494—41no—
253#10623494—41no—
254#10623494—41no—
255#10623494—41no—
256C22×C89—21no—
257C22×C89—21no—
258C22×C89—21no—
259C22×C89C2×C4×D411yescharacteristic
260C2×D89—41no—
261C2×D89—41no—
262C2×D89—41no—
263C4×C89—21no—
264C4×C89—21no—
265C4×C89—21no—
266C4×C89—21no—
267C4×C89C2×#13111yescharacteristic
268C4×C89#56811yescharacteristic
269C4×C89C2×#13711yescharacteristic
270C4×C89#57511yescharacteristic
271Dic89—22no—
272Dic89—41no—
273Dic89—41no—
274Dic89—41no—
275Dic89—41no—
276Dic89—41no—
277Dic89—41no—
278#18013—891nomaximal, Sylow
279C2×C89—41no—
280C2×C89#293311yescharacteristic
281C2×C89C4×#13711yescharacteristic
282C2×C89#295711yescharacteristic
283D89—81no—
284#3107—891no—
285#2971—891no—
286#2962—891no—
287#1592—891no—
288#1590—891no—
289#1588—891no—
290#1587—891no—
291#1584—891no—
292#1583—891no—
293#1579—891no—
294C8×Q16—891no—
295#1573—891no—
296#1497—891no—
297#1477—891no—
298#1453D8911yescharacteristic, complemented
299C89#1801311yescharacteristic, complemented, Sylow
300#533—891no—
301C4×#41—891no—
302#485—891no—
303#485—891no—
304#475—891no—
305#474—891no—
306#462—891no—
307#461—891no—
308#452—891no—
309#452—891no—
310#446—891no—
311#446—891no—
312#446—1781no—
313#446—1781no—
314C8×Q8—891no—
315C8×Q8—891no—
316C8×Q8—891no—
317C8×Q8—891no—
318C4×Q16—891no—
319C4×Q16—891no—
320C4×SD16—891no—
321C4×SD16—891no—
322#436—891no—
323#436—891no—
324#436—1781no—
325#436—1781no—
326#432—891no—
327#432—1781no—
328#431C2×D8911yescharacteristic
329#431C2×D8911yescharacteristic
330#424—1781no—
331C2×C4⋊C8—891no—
332#405—1781no—
333C4×M16—1781no—
334C4×M16C2×D8911yescharacteristic
335#335—891no—
336#335—891no—
337#334—891no—
338#334—891no—
339Q8⋊C8—891no—
340Q8⋊C8—891no—
341Q8⋊C8—891no—
342Q8⋊C8—891no—
343Q8⋊C8—891no—
344Q8⋊C8—891no—
345D4⋊C8—891no—
346D4⋊C8—891no—
347#322C2×D8911yescharacteristic
348#322C2×D8911yescharacteristic
349#322C2×D8911yescharacteristic
350C82C2×D8911yescharacteristic
351C2×#41—891no—
352C2×Q16—891no—
353C2×Q16—891no—
354C2×SD16—891no—
355C2×SD16—891no—
356C2×M16—1781no—
357C2×M16—1781no—
358C2×M16D35611yescharacteristic
359C2×M16D35611yescharacteristic
360C22×C8—1781no—
361C4⋊Q8—891no—
362C4⋊Q8—891no—
363#125—1781no—
364C22⋊Q8—891no—
365C22⋊Q8—891no—
366C22⋊Q8—1781no—
367C4×Q8—891no—
368C4×Q8—891no—
369C4×Q8—891no—
370C4×Q8—891no—
371C4×Q8—1781no—
372C4×D4—891no—
373C4×D4—891no—
374C4×D4—1781no—
375C4×D4—1781no—
376#117—891no—
377#117—1781no—
378#117—1781no—
379C2×C42—1781no—
380C2×C42C22×D8911yescharacteristic
381#107—891no—
382#107—891no—
383#106—891no—
384#106—891no—
385C4⋊C8—21no—
386C4⋊C8—891no—
387C4⋊C8—891no—
388C4⋊C8—891no—
389C4⋊C8—891no—
390C4⋊C8—1781no—
391C4⋊C8—1781no—
392C4⋊C8—1781no—
393C4⋊C8—1781no—
394C4⋊C8—1781no—
395C4⋊C8—1781no—
396C4⋊C8—1781no—
397C4⋊C8C4×D8911yescharacteristic
398C4⋊C8C4×D8911yescharacteristic
399#103—891no—
400#103—891no—
401#103—891no—
402#103—891no—
403#103—891no—
404#103—891no—
405#102—891no—
406#102—891no—
407C22⋊C8—21no—
408C22⋊C8—1781no—
409C22⋊C8—1781no—
410C22⋊C8—1781no—
411C22⋊C8C4×D8911yescharacteristic
412C22⋊C8C4×D8911yescharacteristic
413#97—1781no—
414#97—1781no—
415#97—1781no—
416#97D35611yescharacteristic
417#97D35611yescharacteristic
418C4×C8—1781no—
419C4×C8—1781no—
420C4×C8—1781no—
421C4×C8—1781no—
422C4×C8C22×D8911yescharacteristic
423C4×C8C22×D8911yescharacteristic
424C4×C8C22×D8911yescharacteristic
425C4×C8C22×D8911yescharacteristic
426C4×C8C22×D8911yescharacteristic
427C4×C8C22×D8911yescharacteristic
428#41—3561no—
429#41—1782no—
430C2×Q8—891no—
431C2×Q8—891no—
432C2×Q8—891no—
433C2×Q8—1781no—
434C2×D4—891no—
435C2×D4—1781no—
436C22×C4—1781no—
437C22×C4—1781no—
438C22×C4—1781no—
439C22×C4C2×C4×D8911yescharacteristic
440C22×C4C2×D35611yescharacteristic
441C22×C4#1308357411yescharacteristic
442Q16—1782no—
443Q16—1782no—
444SD16—3561no—
445SD16—3561no—
446M16—22no—
447M16—41no—
448M16—3562no—
449M16—3562no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—3562no—
453C2×C8—1781no—
454C2×C8—1781no—
455C2×C8—1781no—
456C2×C8—1781no—
457C2×C8C2×C4×D8911yescharacteristic
458C2×C8C2×C4×D8911yescharacteristic
459C2×C8C2×C4×D8911yescharacteristic
460C2×C8C2×C4×D8911yescharacteristic
461C2×C8C2×D35611yescharacteristic
462C2×C8C2×D35611yescharacteristic
463C2×C8C2×D35611yescharacteristic
464C2×C8C2×D35611yescharacteristic
465C4⋊C4—891no—
466C4⋊C4—891no—
467C4⋊C4—891no—
468C4⋊C4—891no—
469C4⋊C4—1781no—
470C4⋊C4—1781no—
471C4⋊C4—1781no—
472C4⋊C4—1781no—
473C4⋊C4—1781no—
474C4⋊C4—1781no—
475C4⋊C4—1781no—
476C4⋊C4—1781no—
477C4⋊C4—1781no—
478C22⋊C4—1781no—
479C22⋊C4—1781no—
480C22⋊C4—1781no—
481C22⋊C4—1781no—
482C22⋊C4—1781no—
483C42—1781no—
484C42—1781no—
485C42—1781no—
486C42—1781no—
487C42—1781no—
488C42C2×C4×D8911yescharacteristic
489C42C2×D35611yescharacteristic
490C42#1308357411yescharacteristic
491C42C23×D8911yescharacteristic
492C23—1781no—
493C23C4×D35611yescharacteristic
494Q8—892no—
495Q8—1781no—
496Q8—1781no—
497Q8—1781no—
498Q8—1781no—
499Q8—1781no—
500Q8—7121no—
501D4—3562no—
502D4—1781no—
503D4—1781no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—3562no—
510C2×C4—3561no—
511C2×C4—3561no—
512C2×C4—1781no—
513C2×C4—1781no—
514C2×C4—1781no—
515C2×C4—1781no—
516C2×C4—1781no—
517C2×C4—1781no—
518C2×C4—1781no—
519C2×C4—7121no—
520C2×C4C4×D35611yescharacteristic
521C2×C4C4×D35611yescharacteristic
522C2×C4C4×D35611yescharacteristic
523C2×C4C22×C4×D8911yescharacteristic
524C2×C4C22×D35611yescharacteristic
525C2×C4C2×#1308357411yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—3561no—
536C8—3561no—
537C8—3561no—
538C8—3561no—
539C8#1566804512yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—3561no—
544C22—3561no—
545C22—3561no—
546C22C2×C4×D35611yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—3561no—
552C4—3561no—
553C4—3561no—
554C4—3561no—
555C4—3561no—
556C4—3561no—
557C4—1782no—
558C4#131×D8911yescharacteristic
559C4#1915236311yescharacteristic
560C4C2×#1566804511yescharacteristic
561C4#1915239811yescharacteristic
562C2—41no—
563C2—7121no—
564C2#2602898011yescharacteristic
565C2C4×#1566804511yescharacteristic
566C2#2602917011yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1453C2
socleC22×C89C2×C4×D4
prime radicalC2×#13083574C2×C4
cosocleC42×C89C24
hypercenter#1453D89
hypocenterC89#18013
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #26029199C2semidirect
4. #26019156C2semidirect
5. #26019144C2semidirect
6. #26019132C2semidirect
7. #26019125C2semidirect
8. #26019104C2semidirect
9. #26019097C2semidirect
10. #26019072C2semidirect
11. C8×Dic356C2semidirect
12. #26019037C2semidirect
13. #26018726C2semidirect
15. C89×#1453C2semidirect
298. #1453D89semidirect
299. C89#18013semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260303242
#260292702
#260291992
#260191562
#260191442
#260191322
#260191252
#260191042
#260190972
#260190722
C8×Dic3562
#260190372
#260187262
#260186402
C89×#14532
#1801389